A158943 -id:A158943 - cp's OEIS frontend

A158945 Triangle read by rows, A158944 * an infinite matrix with A158943 (prefaced with a 1) as the right border: (1, 1, 1, 3, 5, 10, 19, 36, ...) and the rest zeros.

1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 3, 0, 2, 0, 5, 0, 3, 0, 6, 0, 10, 4, 0, 3, 0, 10, 0, 19, 0, 4, 0, 9, 0, 20, 0, 36, 5, 0, 4, 0, 15, 0, 38, 0, 69, 0, 5, 0, 12, 0, 30, 0, 72, 0, 131, 6, 0, 5, 0, 20, 0, 57, 0, 138, 0, 250, 0, 6, 0, 15, 0, 40, 0, 108, 0, 262, 0, 476

Offset: 1

Gary W. Adamson, Mar 31 2009

As a property of eigentriangles, sum of n-th row terms = rightmost term of next row. Right border = A158943 prefaced with a 1: (1, 1, 1, 3, 5, 10, 19, 36, 69, ...).

			First few rows of the triangle:
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0,  3;
  3, 0, 2,  0,  5;
  0, 3, 0,  6,  0, 10;
  4, 0, 3,  0, 10,  0, 19;
  0, 4, 0,  9,  0, 20,  0,  36;
  5, 0, 4,  0, 15,  0, 38,   0,  69;
  0, 5, 0, 12,  0, 30,  0,  72,   0, 131;
  6, 0, 5,  0, 20,  0, 57,   0, 138,   0, 250;
  0, 6, 0, 15,  0, 40,  0, 108,   0, 262,   0, 476;
  7, 0, 6,  0, 25,  0, 76,   0, 207,   0, 500,   0, 907;
  ...
Row 5 = (3, 0, 2, 0, 5) = termwise products of (3, 0, 2, 0, 1) and (1, 1, 1, 3, 5); where (3, 0, 2, 0, 1) = row 5 of triangle A158944.

Cf. A158943, A158944.

Triangle read by rows, A158944 * an infinite matrix with A158943 (prefaced with a 1) as the right border: (1, 1, 1, 3, 5, 10, 19, 36, ...) and the rest zeros.

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 50, 95, 181, 345, 657, 1252, 2385, 4544, 8657, 16493, 31422, 59864, 114051, 217286, 413966, 788674, 1502555, 2862617, 5453761, 10390321, 19795288, 37713313, 71850128, 136886433, 260791401, 496850954, 946583628

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = number of compositions of n with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1.1 of the Hoggatt et al. reference. Example: a(4)= 7 because we have 22, 31, 13, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016.
Diagonal sums of A054142. - Paul Barry, Jan 21 2005
Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 27 2009
Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - M. Poyraz Torcuk, Dec 10 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
Todd Mullen, Richard Nowakowski, and Danielle Cox, Counting Path Configurations in Parallel Diffusion, arXiv:2010.04750 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A275446.
Bisection of A003269 (odd part),
Cf. A236582, A251074, A236580.

a:=[1,1,2,4];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, May 09 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}]  (* Michael De Vlieger, Aug 17 2016 *)
LinearRecurrence[{1,2,0,-1},{1,1,2,4},40] (* Harvey P. Dale, Apr 12 2018 *)

my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ G. C. Greubel, May 09 2019

((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
a(n) = a(n-1) + 2*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4.
a(n) = Sum_{alpha = RootOf(1-x-2*x^2+x^4)} (1/283)*(27 + 112*alpha + 9*alpha^2 -48*alpha^3)*alpha^(-n-1).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - Paul Barry, Jan 21 2005
a(n) = A158943(n) -A158943(n-2). - R. J. Mathar, Jan 13 2023

More terms from James Sellers, Jun 05 2000

A297299 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 5, 5, 1, 1, 1, 6, 9, 10, 8, 1, 1, 1, 9, 17, 21, 19, 13, 1, 1, 1, 13, 32, 50, 49, 36, 21, 1, 1, 1, 19, 60, 130, 157, 114, 69, 34, 1, 1, 1, 28, 113, 332, 600, 495, 266, 131, 55, 1, 1, 1, 41, 213, 840, 2161, 2816, 1574, 620, 250, 89, 1, 1, 1

Offset: 1

R. H. Hardin, Dec 27 2017

Table starts
.1.1..1...1....1.....1......1.......1........1.........1..........1...........1
.1.1..2...3....4.....6......9......13.......19........28.........41..........60
.1.1..3...5....9....17.....32......60......113.......213........401.........755
.1.1..5..10...21....50....130.....332......840......2128.......5408.......13772
.1.1..8..19...49...157....600....2161.....7479.....25639......88307......306281
.1.1.13..36..114...495...2816...14138....64406....288079....1310914.....6040052
.1.1.21..69..266..1574..13504...93906...551477...3159693...18792129...113862870
.1.1.34.131..620..5031..65521..629563..4738693..34652662..269300602..2142898994
.1.1.55.250.1446.16123.319835.4235806.40753529.379882818.3857849537.40317718384

			Some solutions for n=5 k=4
..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..1. .0..1..1..0
..0..1..1..0. .0..0..0..0. .0..0..0..0. .0..1..1..0. .1..1..0..0
..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0
..0..0..1..1. .0..1..1..0. .0..1..1..0. .0..1..1..0. .0..1..1..0
..0..1..1..0. .0..0..0..0. .1..1..0..0. .0..0..0..0. .1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..680

Column 3 is A000045(n+1).
Column 4 is A158943(n+1).
Row 2 is A000930.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1)
k=3: a(n) = a(n-1) +a(n-2)
k=4: a(n) = a(n-1) +2*a(n-2) -a(n-4)
k=5: a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -2*a(n-4) for n>5
k=6: a(n) = 4*a(n-1) -8*a(n-3) -2*a(n-4) +4*a(n-5) +4*a(n-6) +a(n-7) -a(n-8)
k=7: [order 12] for n>14
Empirical for row n:
n=1: a(n) = a(n-1)
n=2: a(n) = a(n-1) +a(n-3)
n=3: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4) for n>6
n=4: [order 13] for n>16
n=5: [order 32] for n>37
n=6: [order 68] for n>76

A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 3, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1

Offset: 1

Gary W. Adamson, Mar 31 2009

Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)

			First few rows of the triangle =
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  3, 0, 2, 0, 1;
  0, 3, 0, 2, 0, 1;
  4, 0, 3, 0, 2, 0, 1;
  0, 4, 0, 3, 0, 2, 0, 1;
  5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   1,  0, -2,  0,  1;
   0,  1,  0, -2,  0,  1;
   0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  0,  1,  0, -2,  0,  1;
   ... - _Peter Bala_, Aug 15 2021

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A158943, A158945, A156663.

seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
From Peter Bala, Aug 15 2021: (Start)
T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

A373653 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-5k-1,k).

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 34, 71, 153, 322, 686, 1455, 3088, 6558, 13917, 29548, 62721, 133146, 282646, 599998, 1273690, 2703794, 5739647, 12184181, 25864698, 54905857, 116554700, 247423522, 525233175, 1114970351, 2366870474, 5024416818, 10665883415, 22641646338

Offset: 0

Seiichi Manyama, Jun 12 2024

Index entries for linear recurrences with constant coefficients, signature (1,3,0,-3,0,1).

Cf. A158943, A373638.

a(n) = sum(k=0, n\2, binomial(3*n-5*k-1, k));

G.f.: 1 / (1 - x/(1 - x^2)^3).

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 6.

Showing 1-6 of 6 results.

cp's OEIS Frontend

A158945 Triangle read by rows, A158944 * an infinite matrix with A158943 (prefaced with a 1) as the right border: (1, 1, 1, 3, 5, 10, 19, 36, ...) and the rest zeros.

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A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).

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A297299 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

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Maple

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A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A373653 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k-1,k).

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Formula

A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4).

A373653 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-5k-1,k).